Why can I not select any vertices at all?
This is weird. Check the following:


The selected Object is a Mesh (not a Lamp, Camera, Empty or other form of weirdom).
You are in Edit Mode. To check this, look at the lower part of the 3D-View for selectable box with the text Edit Mode or Object Mode. If you hit Tab (with your cursor in the 3D-View proper), it should switch between Edit- and Object Mode.
Hit A to select / deselect all vertices. 
IF no visible change happens, you are doing art. Else, it is the vertexial selection that fails. 


Hope this helps.